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Change of base

A logarithm in one base can be rewritten as a fraction of logarithms in another base. The numerator is the logarithm of the argument, and the denominator is the logarithm of the old base. The formula applies both to the common logarithm and to the natural logarithm. It is used when the calculator does not have the base the task asks for. The answer is the exponent the original base must have.

\log_b a = \frac{\ln a}{\ln b}

change of base

b^n = a \Rightarrow n = \log_b a

the exponent

Symbols

$b$	the base
$a$	the argument

Example

$\log_2 32 = \dfrac{\ln 32}{\ln 2} = 5$ , because $2^5 = 32$ .

The denominator is the logarithm of the base, not of the argument.

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Part of Mathematics S1: Logarithms and exponential equations.